Question: Yusef earned a total of $180$ dollars from drawing $12$ identically-priced pictures at a festival. Write an equation to describe the relationship between $p$, the number of pictures, and $e$, the total earnings.
Answer: Let's find the constant of proportionality. In the proportional relationship between $p$, the number of pictures, and $e$, the total earnings, one constant of proportionality is the price of each picture. It is the number we multiply by the number of pictures to get the total earnings. $p\,\times\, ?=e$ $\begin{aligned} p\,\times\, {?}&=e \\\\ {?}&=\dfrac{e}{p} \\\\ &=\dfrac{180}{12} \\\\ &={15} \end{aligned}$ The constant of proportionality is ${15}$. This means we can multiply ${15}$ by the number of pictures to get the total earnings. Now, let's write the equation: $\begin{aligned} \text{total earnings}&={\text{price per picture}}\times\text{number of pictures} \\\\ e&={15}p \end{aligned}$ One correct equation is: $e=15p$